Coplanar Waveguide (CPW) Calculator

Quasi-static conformal-mapping analysis (Ghione & Naldi 1987). Valid when W, S ≪ λ.

Relative dielectric constant ε_r: Trace width W [μm]: Gap width S [μm]: Substrate thickness h [μm]: Back-side ground plane: Calculate

Formulas

Quantity	Formula
Geometry modulus	k = W / (W + 2S)    k′ = √(1 − k²)
Substrate factor k1  (conventional)	k1 = sinh(πW / 4h) / sinh(π(W + 2S) / 4h)
Substrate factor k2  (CBCPW)	k2 = tanh(πW / 4h) / tanh(π(W + 2S) / 4h)
εeff  (conventional CPW)	1 + (εr − 1) / 2  ×  [K(k1) / K(k1′)] / [K(k) / K(k′)]
εeff  (CBCPW)	[K(k)/K(k′) + εr K(k2)/K(k2′)] / [K(k)/K(k′) + K(k2)/K(k2′)]
Z0  (conventional CPW)	(30π / √εeff) × K(k′) / K(k)
Z0  (CBCPW)	60π / [√εeff × (K(k)/K(k′) + K(k2)/K(k2′))]
K(k)	Complete elliptic integral of the first kind — computed via arithmetic-geometric mean (AGM)
Phase velocity & guided wavelength	vph = c / √εeff    λg = c / (f √εeff)
What is εeff?	The effective permittivity experienced by the propagating mode. CPW fields extend into both the substrate (εr) and the air (1), so εeff is a weighted average between them. It determines phase velocity and guided wavelength. For a thick substrate (W, S ≪ h): εeff → (1 + εr) / 2. For a thin substrate without back GND: εeff → 1. For a thin substrate with back GND: εeff → εr.